A medical may be either...
1) In case it is not that serious, that you don't need to divert, I would call PAN too. Like taking place at the end of a flight, PAN call to have medical personal available after arriving. Or maybe even it can be handled directly with the handling agent, without informing ATC.
2) A passenger is having a heart attack and DYING and a matter of minutes of touching down may make the difference between life and death:
a) Distress: a condition of being threatened by serious and/or IMMINENT danger and of requiring IMMEDIATE assistance.
= MAYDAY
You are making a classical mistake and mixing up phases of an emergency (uncertainty, alert and distress phases), with when a pilot should use emergency communication and which. Those 2 things are not connected. These phases has not necessarily to do with how ATC will handle emergency traffic, but rather the things surrounding it (when to launch SAR, roll fire brigade etc.).